On $2$-pyramidal Hamiltonian cycle systems

Abstract

A Hamiltonian cycle system of the complete graph minus a $1$--factor $K_{2v}-I$ (briefly, an HCS$(2v)$) is {\it $2$-pyramidal} if it admits an automorphism group of order $2v-2$ fixing two vertices. In spite of the fact that the very first example of an HCS$(2v)$ is very old and 2-pyramidal, a thorough investigation of this class of HCSs is lacking. We give first evidence that there is a strong relationship between 2-pyramidal HCS$(2v)$ and {\it$1$-rotational} Hamiltonian cycle systems of the complete graph $K_{2v-1}$. Then, as main result, we determine the full automorphism group of every 2-pyramidal HCS$(2v)$. This allows us to obtain an exponential lower bound on the number of non-isomorphic $2$-pyramidal HCS$(2v)$.